E^2=(mc^2)^2+(pc)^2

$E^{2}=(m_{0}c^{2})^{2}+(pc)^{2}$ is a formula in special relativity that relates the relativistic energy, E, rest mass, $m_{0}$, and momentum, p, of a particle. From it the momentum of a photon can be derived and so to the famous equation E=mc².

Derivation

The relativistic equation for momentum is:

$p=\gamma m_{0}v$

The equation for relativistic energy is:

$E=\gamma m_{0}c^{2}$

These can be rearranged into the forms: $({\frac {E}{m_{0}c^{2}}})^{2}=\gamma ^{2}$
$({\frac {p}{m_{0}c}})^{2}=({\frac {\gamma v}{c}})^{2}$

Subtracting the momentum equation from the energy equation an rearranging gives: $E^{2}=(m_{0}c^{2})^{2}+(pc)^{2}$

Other Formulae

The energy of a photon can be derived by setting the rest mass, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle m_0} , equal to zero, so: Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle E=pc}

Mass energy equivalence can be derived by setting the momentum, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p} , to zero. This gives us one of the most famous equations in physics, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle E=m_0 c^2}